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Dash through a series of danger filled levels and save the world from evil in the Geometry Dash World universe. Each level has a new adventure and peril, requiring players to control their character to complete each level. In some levels, players must dash through gaps and spikes that shoot out of the air. In this game, your primary goal is to outmaneuver the many lethal obstacles that are set up to stop you. If you can deftly avoid each death trap, you will be able to proceed without much difficulty. While traveling on the road, you must be very careful to pause between each step. Doing this will keep your character moving throughout the level without stopping. All you have to do is press a button at the right time to complete the task. Be mindful of obstacles and detours while playing the game; they may lead to a change in direction. If you get stuck or hit an obstacle, you must restart the game from the beginning.
A number of processing methods for cycle slip have been studied over the years. In particular, the cycle-slip detection and repair method using dual-frequency observations was proposed by Blewitt [10], Gao and Li [11], Bisnath et al. [7], Liu [12], and Cai et al. [13]. These popular methods employ two complementary geometry-free linear combinations for dynamic users, the Melbourne-Wübbena (MW) combination and ionosphere combination, since the combined cycle slip can be close to zero and cannot be detected using only one combination. For example, the same size of cycle slips on L1 and L2 frequencies such as (10, 10) cannot be noticed on the MW combination, but it has a detectable value on the ionosphere combination. Similarly, the special cycle-slip pair (77, 60), which does not influence the ionosphere combination, is detected by the MW combination. We call these special cycle-slip pairs insensitive pairs [4,13,14].
Most dual-frequency insensitive cycle-slip pairs can be detected by the MW combination and ionosphere combination. In addition, this detection algorithm can be easily used for static and even dynamic users because of the benefit of geometry-free combination; however, it has critical limitations when detecting small cycle slips. First, these techniques are affected by noise and pseudorange multipath effects even if a smoothing or averaging technique such as a Hatch filter and low-pass filter is applied. In other words, these techniques cannot be processed instantaneously and might fail to detect small cycle slips under high-multipath conditions [13,14]. Furthermore, the ionosphere combination is vulnerable to severe ionospheric storm since the remaining bias of the ionospheric variation makes it difficult to detect small cycle slips [13,15,16]. The probability of occurrence of these insensitive pairs is incredibly small [10], but it is still considered an integrity threat to a system requiring high-integrity performance.
Considering the limitations discussed above, we propose a novel algorithm for cycle-slip detection and compensation using only the dual-frequency carrier-phase observations without pseudorange. In order for the network RTK to be used as safety-critical systems in the near future, it is necessary to ensure a high-integrity information as well as a high-precision corrections. As a first step to enhancing the network-based RTK system, we focus on the high-integrity detection algorithm for continuously operating reference stations (CORS). In this paper, we introduce two independent and complementary ionosphere combinations: one is the geometry-free ionosphere combination usually used in cycle-slip detection, and the other is a new geometry-based ionosphere combination to replace the MW combination. We name these combinations the ionospheric negative and positive combination, respectively. Song and Kee demonstrated that there is no additional geometry-free carrier-phase combination to replace the MW combination for dynamic users [14]; however, geometry-based combinations can be employed in static permanent stations for generating reliable high-precision corrections. Because our ionospheric positive combination constructs the combined cycle slips with positive signs, it can detect a small cycle slip more efficiently compared to the popularly used geometry-based ionosphere-free combination [23,24]. Furthermore, we apply a second-order time-differenced observation (or acceleration) to reduce the influence of ionospheric delay under severe ionospheric storm regardless of baseline lengths [13,15,25].
We call Equation (6) the TDSD carrier-phase residual. In this section, we describe two L1/L2 linear combinations of carrier phase for obtaining the MVs of cycle-slip detection. The first observation is the geometry-free combination that only contains ionospheric slant delay, which is widely used to detect dual-frequency cycle slips [7,10,11,12,13,25,41]. In this paper, we named it the ionosphere negative (IN) combination since L1 and L2 cycle slips are combined with negative signs:
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